On instantons as Kaluza-Klein modes of M5-branes

Abstract

Instantons and W-bosons in 5d maximally supersymmetric Yang-Mills theory arise from a circle compactification of the 6d (2,0) theory as Kaluza-Klein modes and winding self-dual strings, respectively. We study an index which counts BPS instantons with electric charges in Coulomb and symmetric phases. We first prove the existence of unique threshold bound state of (noncommutative) U (1) instantons for any instanton number, and also show that charged instantons in the Coulomb phase correctly give the degeneracy of SU (2) self-dual strings. By studying SU (N ) self-dual strings in the Coulomb phase, we find novel momentum-carrying degrees on the worldsheet. The total number of these degrees equals the anomaly coefficient of SU (N ) (2,0) theory. We finally show that our index can be used to study the symmetric phase of this theory, and provide an interpretation as the superconformal index of the sigma model on instanton moduli space.

From strings in 6d to strings in 5d

“…In doing this we connect to recent mathematical results on generating functions of Poincaré polynomials of moduli spaces of sheaves on ruled surfaces [15]. Perturbative expansions of elliptic genera for strings of the 5d N = 1 * theory have been obtained in the past using instanton calculus [16], but our results are more complete since they provide analytic expressions for elliptic genera of any magnetic charge.…”

“…The mass-deformation breaks the leftmoving supersymmetry and can be thought of as introducing a grading in the cohomology ring of the moduli space of the string given by powers of Q m . In order to define an elliptic genus for the (4, 4) string reference [16] proposed to set Q m = −1. However, recalling that the mass m couples to the R-symmetry current J I , one can extract from (4.10) the more refined information (see also page 4 of [30] for an explanation of the relation between Hodge numbers, spin and R-symmetry) shown in tables 2, 3 and 4.…”

Little strings and T-duality

“…This can be computed as follows. (See [22] for a very similar analysis of the perturbative partition function of 5d maximal SYM.) In the Coulomb phase, both JHEP02(2016)170 momentum along the circle and the electric charge preserve supercharges Q aα+ and Q Aα − , and breaks Qα a+ and Q A α− .…”

“…See e.g. [22] for more details and illustrations. This Young diagram expression first appeared in the instanton calculus of the 4d and 5d SYM [23,24], and our result (2.7) is simply an elliptic uplift of their results.…”

“…Therefore, the single particle index of the perturbative particles always contains the following overall factor coming from the spacetime 0-modes [22],…”